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The knapsack cryptosystem at work

This note is an example of the knapsack crypto-system at work.So before you got through this note I recommend reading the main one . Click here

Suppose that a typical user of this crypto-system selects as the secret key the super-increasing sequence \({3,5,11,20,41}\) and the modulus \(m = 85\) and the multiplier \(a = 44\).Each member of the super-increasing sequence is multiplied by 44 and reduced modulo 85 to yield \({47,50,59,30,19}\).This is the encryption key the user submit to the public directory.

Suppose someone wants to send a plaintext message to the user such as "HELP US". First we convert it into a string of 1's and 0's \[ M = 00111 00100 01011 10100 10010\] The string is then broken up into blocks of digits,in the current case,blocks of length 5.Using the listed public keys to encrypt,the sender transforms the successive blocks into \[47*0+50*0+59*1+30*1+19*1 = 108\] \[47*0+50*0+59*1+30*0+19*0 = 59\] \[47*0+50*1+59*0+30*1+19*1 = 99\] \[47*0+50*1+59*1+30*1+19*1 = 158\] \[47*1+50*0+59*1+30*0+19*0 = 106\] \[47*1+50*0+59*0+30*1+19*0 = 77\]

The transmitted ciphertext consists of the following series of positive integers \({108,59,99,158,106,777}\).

To read the message the legitimate receiver first solves the congruence \[44x \equiv 1 \pmod{85}\],yielding \[y \equiv 29 \pmod{85}\].Then each ciphertext number is multiplied by 29 and reduced modulo 85 to produce a super-increassing knapsack problem.

For example,\(108\) is converted to \(72\) as \[108*29 \equiv 72 \pmod{85}\] and the corresponding knapsack problem is \[3x_1 + 5x_2 + 11x_3 + 20x_4+41x_5 = 72\].

The procedure for solving super-increasing knapsack problems quickly yields the solution \(x_1 = x_2 =0\),\(x_3 = x_4=x_5 = 1\),in this way \(00111\),the first block of the binary equivalent of the plaintext is obtained.

Note by Eddie The Head
3 years, 1 month ago

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